Data from the U.S. Department of Health and Human Services...

Data from the U.S. Department of Health and Human Services indicates that the cumulative number $N$ of reported cases of AIDS in the United States in year $x$ can be approximated by the equation $$ N=3362.1 x^{2}-17,270.3 x+24,043 $$ where $x=0$ corresponds to $1980 .$ In what year did the total reach $550,000 ?$

Data from the U.S. Department of Health and Human Services indicates that the cumulative number $N$ of reported cases of AIDS in the United States in year $x$ can be approximated by the equation
$$
N=3362.1 x^{2}-17,270.3 x+24,043
$$
where $x=0$ corresponds to $1980 .$ In what year did the total reach $550,000 ?$

Key Concepts

Quadratic Functions

Quadratic functions are mathematical expressions of the form ax² + bx + c, where the highest power of the variable is 2. They are widely used in various fields for modeling relationships where the rate of change itself changes linearly, and their graphs are parabolic in shape. This concept forms the basis for understanding how a situation can be represented by such an equation.

Solving Quadratic Equations

Solving quadratic equations involves finding the values of the variable that satisfy the equation. Techniques for solving include factoring, completing the square, and applying the quadratic formula. This process is crucial for determining the specific points at which a modeled scenario reaches a known outcome.

Mathematical Modeling

Mathematical modeling refers to the process of representing real-world phenomena using mathematical expressions and equations. By assigning variables and constants in the model, real-world data can be analyzed and predictions can be made. This concept is essential for translating empirical data into a mathematical framework.

Interpreting Solutions in Context

Interpreting solutions in context involves relating the numerical or algebraic outcomes of an equation back to the real-world scenario being modeled. It requires understanding the significance of computed values and ensuring they are meaningful within the frame of the original problem, such as converting a computed variable into a specific calendar year.

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